Subgroups — #30641137

Order 16256.

656 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861168	C₂	1	2	yes	maximal
2	#22549746	C₂	1	2	yes	maximal, complemented
3	#22549742	C₂	1	2	yes	maximal, complemented
4	#22549736	C₂	1	4	yes	maximal
5	C₄×C₄⋊Dic₁₂₇	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	#16861168	—	2	8	no	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	8	yes	—
10	C₂×C₂²⋊Dic₁₂₇	C₂²	1	4	yes	—
11	C₂×C₄×Dic₁₂₇	C₂²	1	4	yes	—
12	C₁₂₇⋊#95	C₂²	1	8	yes	—
13	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₂₇	C₄	1	8	yes	complemented
16	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₂₇	C₂³	1	8	yes	—
20	C₄⋊Dic₁₂₇	—	2	8	no	—
21	C₄⋊Dic₁₂₇	—	2	16	no	—
22	C₂²⋊Dic₁₂₇	—	2	8	no	—
23	C₄×Dic₁₂₇	—	2	8	no	—
24	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	—	2	8	no	—
26	C₂²×C₄×C₁₂₇	C₂×C₄	1	8	yes	—
27	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
29	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
30	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
35	C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
36	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
37	C₂×Dic₁₂₇	—	2	8	no	—
38	C₂×Dic₁₂₇	—	2	16	no	—
39	C₂³×C₁₂₇	—	2	4	no	—
40	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
41	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
42	C₂³×C₁₂₇	C₂×Q₈	1	2	yes	—
43	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	—	2	4	no	—
45	C₂×C₄×C₁₂₇	—	2	16	no	—
46	C₂×C₄×C₁₂₇	C₄⋊C₄	1	16	yes	complemented
47	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₁₂₇	#41	1	8	yes	—
56	C₂²×C₁₂₇	—	2	4	no	—
57	C₂²×C₁₂₇	—	2	4	no	—
58	C₂²×C₁₂₇	—	2	4	no	—
59	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
60	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
61	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
62	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
63	C₂²×C₁₂₇	C₂×#41	1	4	yes	—
64	C₄×C₁₂₇	—	2	4	no	—
65	C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
66	C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
67	C₄×C₁₂₇	#117	1	8	yes	complemented
68	Dic₁₂₇	—	4	8	no	—
69	C₂×C₁₂₇	—	2	4	no	—
70	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₁₂₇	C₂×#117	1	2	yes	—
72	C₂×C₁₂₇	#532	1	2	yes	—
73	C₂×C₁₂₇	#533	1	2	yes	—
74	#2285	—	127	1	no	maximal, Sylow
75	C₁₂₇	#2285	1	1	yes	characteristic, complemented, Sylow
76	C₂×#117	—	127	2	no	—
77	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic
78	#384	—	127	2	no	—
79	#382	—	127	2	no	—
80	#380	—	127	4	no	—
81	C₄×C₄⋊C₄	—	127	4	no	—
82	C₂³×C₄	C₂×D₁₂₇	1	2	yes	—
83	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
84	#117	—	254	8	no	—
85	C₂×C₄⋊C₄	—	127	4	no	—
86	C₂×C₄⋊C₄	—	127	8	no	—
87	C₂×C₂²⋊C₄	—	127	4	no	—
88	C₂×C₄²	—	127	4	no	—
89	C₂×C₄²	Dic₁₂₇	1	8	yes	complemented
90	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
93	#95	—	127	8	no	—
94	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
95	C₂²×C₄	—	2	8	no	—
96	C₂²×C₄	—	127	8	no	—
97	C₂²×C₄	C₂×Dic₁₂₇	1	8	yes	—
98	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
99	C₂²×C₄	D₅₀₈	1	4	yes	—
100	C₂²×C₄	Dic₂₅₄	1	4	yes	—
101	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
102	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
105	C₄⋊C₄	—	254	8	no	—
106	C₄⋊C₄	—	254	16	no	—
107	C₂²⋊C₄	—	254	8	no	—
108	C₄²	—	254	8	no	—
109	C₄²	C₂×Dic₁₂₇	1	4	yes	—
110	C₄²	C₂×Dic₁₂₇	1	8	yes	—
111	C₄²	C₂×Dic₁₂₇	1	4	yes	—
112	C₂³	—	2	4	no	—
113	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
114	C₂³	C₂×D₅₀₈	1	2	yes	—
115	C₂³	C₂×Dic₂₅₄	1	2	yes	—
116	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	16	no	—
119	C₂×C₄	—	254	8	no	—
120	C₂×C₄	—	254	16	no	—
121	C₂×C₄	C₄⋊Dic₁₂₇	1	16	yes	complemented
122	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
123	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
124	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
125	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
126	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
127	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
128	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
129	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
130	C₂×C₄	#14523248	1	8	yes	—
131	C₂²	—	2	4	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	2	4	no	—
134	C₂²	C₂×C₄⋊Dic₁₂₇	1	4	yes	—
135	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
136	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
137	C₂²	C₂²×Dic₂₅₄	1	1	yes	characteristic
138	C₂²	C₂×#14523248	1	4	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	508	8	no	—
141	C₄	C₂×C₄⋊Dic₁₂₇	1	4	yes	—
142	C₄	C₂×C₄⋊Dic₁₂₇	1	4	yes	—
143	C₄	#16861168	1	8	yes	complemented
144	C₂	—	2	4	no	—
145	C₂	C₂²×C₄⋊Dic₁₂₇	1	1	yes	characteristic
146	C₂	C₂×#16861168	1	2	yes	—
147	C₂	#22550426	1	2	yes	—
148	C₂	#22550433	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂³	C₂³×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle, cosocle	C₂³×C₁₂₇	C₂⁴
prime radical	C₂⁴×C₁₂₇	C₂³
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	#2285
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #22549746	C₂	semidirect
3. #22549742	C₂	semidirect
5. C₄×C₄⋊Dic₁₂₇	C₂	semidirect
15. C₂×C₄²×C₁₂₇	C₄	semidirect
46. C₂×C₄×C₁₂₇	C₄⋊C₄	semidirect
67. C₄×C₁₂₇	#117	semidirect
75. C₁₂₇	#2285	semidirect
89. C₂×C₄²	Dic₁₂₇	semidirect
121. C₂×C₄	C₄⋊Dic₁₂₇	semidirect
143. C₄	#16861168	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16861168	2
#22549746	2
#22549742	2
#22549736	2
C₄×C₄⋊Dic₁₂₇	2
C₂²×C₄²×C₁₂₇	2
#2285	127